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Argyres-Douglas Theories, the Macdonald Index, and an RG Inequality

Matthew Buican, Takahiro Nishinaka

TL;DR

This work develops closed-form Macdonald-limit expressions for the superconformal indices of the Argyres–Douglas theories (A$_1$, A$_{2n-3}$) and (A$_1$, D$_{2n}$) as deformations of Macdonald polynomials, with irregular-singularity wavefunctions as the key ingredient. It provides extensive cross-checks via S-dualities, symmetry enhancements, and RG flows, and shows that HL limits reproduce Higgs-branch Hilbert series, with a precise 3D monopole-number inequality underpinning this equivalence. The authors further relate operator equations beyond the Higgs branch to null states in the associated chiral algebras, and discuss analytic properties of the indices (positivity, pole structure) and implications for broader N=2 SCFTs. The results offer a unified, computable framework for Macdonald refinements of AD indices, linking four-dimensional data to 2d chiral algebras and three-dimensional reductions, and suggesting new constraints on RG flow and operator spectra across dimensions.

Abstract

We conjecture closed-form expressions for the Macdonald limits of the superconformal indices of the (A_1, A_{2n-3}) and (A_1, D_{2n}) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2)_R currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S^1 reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general N=2 superconformal field theories.

Argyres-Douglas Theories, the Macdonald Index, and an RG Inequality

TL;DR

This work develops closed-form Macdonald-limit expressions for the superconformal indices of the Argyres–Douglas theories (A, A) and (A, D) as deformations of Macdonald polynomials, with irregular-singularity wavefunctions as the key ingredient. It provides extensive cross-checks via S-dualities, symmetry enhancements, and RG flows, and shows that HL limits reproduce Higgs-branch Hilbert series, with a precise 3D monopole-number inequality underpinning this equivalence. The authors further relate operator equations beyond the Higgs branch to null states in the associated chiral algebras, and discuss analytic properties of the indices (positivity, pole structure) and implications for broader N=2 SCFTs. The results offer a unified, computable framework for Macdonald refinements of AD indices, linking four-dimensional data to 2d chiral algebras and three-dimensional reductions, and suggesting new constraints on RG flow and operator spectra across dimensions.

Abstract

We conjecture closed-form expressions for the Macdonald limits of the superconformal indices of the (A_1, A_{2n-3}) and (A_1, D_{2n}) Argyres-Douglas (AD) theories in terms of certain simple deformations of Macdonald polynomials. As checks of our conjectures, we demonstrate compatibility with two S-dualities, we show symmetry enhancement for special values of n, and we argue that our expressions encode a non-trivial set of renormalization group flows. Moreover, we demonstrate that, for certain values of n, our conjectures imply simple operator relations involving composites built out of the SU(2)_R currents and flavor symmetry moment maps, and we find a consistent picture in which these relations give rise to certain null states in the corresponding chiral algebras. In addition, we show that the Hall-Littlewood limits of our indices are equivalent to the corresponding Higgs branch Hilbert series. We explain this fact by considering the S^1 reductions of our theories and showing that the equivalence follows from an inequality on monopole quantum numbers whose coefficients are fixed by data of the four-dimensional parent theories. Finally, we comment on the implications of our work for more general N=2 superconformal field theories.

Paper Structure

This paper contains 46 sections, 130 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Introduction
  3. Motivating our Conjectures
  4. Motivating our Conjectures
  5. Low-Rank Checks
  6. Low-Rank Checks
  7. The $(A_1, A_1)$ and $(A_1, D_2)$ theories
  8. The $(A_1, A_1)$ and $(A_1, D_2)$ theories
  9. Symmetry Enhancement in the $(A_1, A_3)$ and $(A_1, D_4)$ theories
  10. Symmetry Enhancement in the $(A_1, A_3)$ and $(A_1, D_4)$ theories
  11. The $(A_3, A_3)$ $S$-duality
  12. The $(A_3, A_3)$ $S$-duality
  13. The $(A_2, A_5)$ $S$-duality
  14. The $(A_2, A_5)$ $S$-duality
  15. General Checks
  16. ...and 31 more sections

Figures (2)

  • Figure 1: The $(A_3,A_3)$ theory can be constructed by gauging a diagonal $SU(2)$ flavor symmetry of two $(A_1,D_4)$ theories and a fundamental hypermultiplet.
  • Figure 2: The $(A_2,A_5)$ theory can be constructed by gauging a diagonal $SU(2)$ flavor symmetry of an $(A_1,A_3)$ sector, an $(A_1, D_6)$ SCFT, and a fundamental hypermultiplet.